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CORRECTION AND COMMENT CONCERNING "ON ... - Project Euclid
G. Leger and E. Luks. Nagoya Math. J. Vol. 59 (1975), 217-218. CORRECTION AND COMMENT CONCERNING. "ON DERIVATIONS AND HOLOMORPHS OF.
G. Leger and E. Luks Nagoya Math. J. Vol. 59 (1975), 217-218
CORRECTION AND COMMENT CONCERNING "ON DERIVATIONS AND HOLOMORPHS OF NILPOTENT LIE ALGEBRAS" G. LEGER AND E. LUKS Nagoya Mathematical Journal Vol. 44, December 1971 pp. 39-50
The examples on p. 44 of metabelian Lie algebras Lγ and L2 are isomorphic over certain base fields contrary to the claim in the paper. More specifically, the error lies in the incorrect statement that the rank of ad x for any x in Lx is either zero or is at least 3. This statement does hold over Q, but if τ is a scalar satisfying τ2 — τ — 1 = 0 then ad (xx + τx2 + τ2x3 + x4 + τ2xδ) has rank 2.
These examples with holomorphs isomorphic to ίf were important in that they were to demonstrate the existence over any field of characteristic 0 of non-isomorphic metabelian Lie algebras with isomorphic derivation algebras and holomorphs. Actually, we have since discovered that there is, up to isomorphism, only one Lie algebra over C whose holomorph is isomorphic to i?. Thus, although Theorem 4.5 is technically still justified by the examples, we feel it important now to point out that there do exist metabelian Lie algebras with isomorphic derivation algebras and holomorphs which are nevertheless non-isomorphic even under extensions of the base field. We present the following metabelian Lie algebras M19 M2, over any field of characteristic 0; these algebras have isomorphic derivation algebras and have holomorphs isomorphic to H(6,4>: Mj has basis #i, , x^ with the multiplication table
Received November 11, 1974. 217
218
G. LEGER AND E. LUKg
with [xi9 Xj] = 0 for i < j if it is not in the above list. M2 has basis y19 , y10 with the multiplication table [2/1, i/2] = 2/9, [1/1,1/3] = 2/8, [1/1, vJ = 2/10
with [j/i, ^ ] = 0 for i < j iί it is not in the above list. To see that Mx and M2 are non-isomorphic, one may verify that the rank of ad x for x in Mγ is either 0 or at least 3 (and this is valid for all fields) while in M2 the derivations ad yi9 ad y6 and ad y6 each have rank 2. Tufts University Bucknell University
